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% $Header: /u/u3/gcmpack/manual/part6/exch2.tex,v 1.9 2004/03/12 20:58:19 afe Exp $ |
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%% * Introduction |
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%% o what it does, citations (refs go into mitgcm_manual.bib, |
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%% preferably in alphabetic order) |
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%% o Equations |
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%% * Key subroutines and parameters |
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\section{exch2: Extended Cubed Sphere \mbox{Topology}} |
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\label{sec:exch2} |
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\subsection{Introduction} |
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The \texttt{exch2} package is an extension to the original cubed |
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sphere topological configuration that allows more flexible domain |
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decomposition and parallelization. Cube faces (also called |
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subdomains) may be divided into any number of tiles that divide evenly |
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into the grid point dimensions of the subdomain. Furthermore, the |
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individual tiles may be run on separate processors in different |
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combinations, and whether exchanges between particular tiles occur |
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between different processors is determined at runtime. This |
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flexibility provides for manual compile-time load balancing across a |
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relatively arbitrary number of processors. \\ |
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The exchange parameters are declared in |
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\filelink{pkg/exch2/W2\_EXCH2\_TOPOLOGY.h}{pkg-exch2-W2_EXCH2_TOPOLOGY.h} |
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and assigned in |
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\filelink{pkg/exch2/w2\_e2setup.F}{pkg-exch2-w2_e2setup.F}. The |
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validity of the cube topology depends on the \texttt{SIZE.h} file as |
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detailed below. Both files are generated by Matlab scripts in |
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\texttt{utils/exch2/matlab-topology-generator}; see Section |
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\ref{sec:topogen} for details on creating alternate topologies. The |
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default files provided in the release configure a cubed sphere |
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topology of six tiles, one per subdomain, each with 32$\times$32 grid |
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points, all running on a single processor. Pregenerated examples of |
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these files with alternate topologies are provided under |
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\texttt{utils/exch2/code-mods} along with the appropriate |
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\texttt{SIZE.h} file for single-processor execution. |
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\subsection{Invoking exch2} |
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To use exch2 with the cubed sphere, the following conditions must be |
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met: \\ |
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$\bullet$ The exch2 package is included when \texttt{genmake2} is run. The |
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easiest way to do this is to add the line \texttt{exch2} to the |
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\texttt{profile.conf} file -- see Section \ref{sect:buildingCode} |
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for general details. \\ |
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$\bullet$ An example of \texttt{W2\_EXCH2\_TOPOLOGY.h} and |
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\texttt{w2\_e2setup.F} must reside in a directory containing code |
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linked when \texttt{genmake2} runs. The safest place to put these |
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is the directory indicated in the \texttt{-mods=DIR} command line |
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modifier (typically \texttt{../code}), or the build directory. The |
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default versions of these files reside in \texttt{pkg/exch2} and are |
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linked automatically if no other versions exist elsewhere in the |
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link path, but they should be left untouched to avoid breaking |
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configurations other than the one you intend to modify.\\ |
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$\bullet$ Files containing grid parameters, named |
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\texttt{tile}???\texttt{.mitgrid} where ??? is \texttt{001} through |
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\texttt{006} (one per subdomain), must be in the working directory |
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when the MITgcm executable is run. These files are provided in the |
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example experiments for cubed sphere configurations with |
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32$\times$32 cube sides and are non-trivial to generate -- please |
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contact MITgcm support if you want to generate files for other |
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configurations. \\ |
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$\bullet$ As always when compiling MITgcm, the file \texttt{SIZE.h} |
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must be placed where \texttt{genmake2} will find it. In particular |
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for the exch2, the domain decompositin specified in \texttt{SIZE.h} |
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must correspond with the particular configuration's topology |
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specified in \texttt{W2\_EXCH2\_TOPOLOGY.h} and |
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\texttt{w2\_e2setup.F}. Domain decomposition issues particular to |
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exch2 are addressed in Section \ref{sec:topogen}: ``Generating |
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Topology Files for exch2''; a more general background on the subject |
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relvant to MITgcm is presented in Section |
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\ref{sect:specifying_a_decomposition}: ``Specifying a |
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decomposition''.\\ |
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As of the time of writing the following examples use exch2 and may be |
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used for guidance: |
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\begin{verbatim} |
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verification/adjust_nlfs.cs-32x32x1 |
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verification/adjustment.cs-32x32x1 |
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verification/aim.5l_cs |
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verification/global_ocean.cs32x15 |
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verification/hs94.cs-32x32x5 |
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\end{verbatim} |
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\subsection{Generating Topology Files for exch2} |
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\label{sec:topogen} |
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Alternate cubed sphere topologies may be created using the Matlab |
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scripts in \texttt{utils/exch2/matlab-topology-generator}. Running the |
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m-file \texttt{driver} from the Matlab prompt (without passing any |
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function parameters) generates exch2 topology files |
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\texttt{W2\_EXCH2\_TOPOLOGY.h} and \texttt{w2\_e2setup.F} in the |
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working directory and displays via Matlab a figure of the topology. |
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The other m-files in the directory are subroutines of \texttt{driver} |
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and should not be run except for development purposes. \\ |
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The parameters that determine the dimensions and topology of the |
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generated configuration are nr, nb, ng, tnx and tny, and all are |
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assigned early in the script. |
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The first three determine the size of the subdomains (cube faces) and |
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hence the size of the overall domain. Each one determines the number |
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of grid points, and therefore the resolution, along the subdomain |
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sides in a ``great circle'' around each axis of the cube. At the time |
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of this writing MITgcm requires these three parameters to be equal, |
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but they provide for future releases of MITgcm to accomodate different |
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resolutions around the axes to allow (for example) greater resolution |
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around the equator.\\ |
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The parameters tnx and tny determine the dimensions of the tiles into |
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which the subdomains are decomposed, and must evenly divide the |
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integer assigned to nr, nb and ng. The result is a rectangular tiling |
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of the subdomain. Figure \ref{fig:24tile} shows one possible topology |
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for a twenty-four tile cube, and figure \ref{fig:12tile} shows one for |
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twelve tiles. \\ |
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\begin{figure} |
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\begin{center} |
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\resizebox{4in}{!}{ |
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\includegraphics{part6/s24t_16x16.ps} |
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} |
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\end{center} |
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\caption{Plot of cubed sphere topology with a 32$\times$32 grid and |
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twenty-four tiles (tnx=16, tny=16) |
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} \label{fig:24tile} |
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\end{figure} |
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\begin{figure} |
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\begin{center} |
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\resizebox{4in}{!}{ |
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\includegraphics{part6/s12t_16x32.ps} |
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} |
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\end{center} |
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\caption{Plot of cubed sphere topology with a 32$\times$32 grid and |
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twelve tiles (tnx=16, tny=32) |
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} \label{fig:12tile} |
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\end{figure} |
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Tiles can be selected from the topology to be omitted from being |
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allocated memory and processors. This kind of tuning is useful in |
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ocean modeling for omitting tiles that fall entirely on land. The |
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tiles omitted are specified in the file \texttt{blanklist.txt} by |
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their tile number in the topology, separated by a newline. \\ |
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\subsection{Key Variables} |
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The descriptions of the variables are divided up into scalars, |
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one-dimensional arrays indexed to the tile number, and two and three |
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dimensional arrays indexed to tile number and neighboring tile. This |
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division actually reflects the functionality of these variables: the |
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scalars are common to every part of the topology, the tile-indexed |
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arrays to individual tiles, and the arrays indexed to tile and |
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neighbor to relationships between tiles and their neighbors. |
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\subsubsection{Scalars} |
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The number of tiles in a particular topology is set with the parameter |
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\texttt{NTILES}, and the maximum number of neighbors of any tiles by |
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\texttt{MAX\_NEIGHBOURS}. These parameters are used for defining the |
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size of the various one and two dimensional arrays that store tile |
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parameters indexed to the tile number.\\ |
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The scalar parameters \varlink{exch2\_domain\_nxt}{exch2_domain_nxt} |
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and \varlink{exch2\_domain\_nyt}{exch2_domain_nyt} express the number |
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of tiles in the x and y global indices. For example, the default |
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setup of six tiles has \texttt{exch2\_domain\_nxt=6} and |
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\texttt{exch2\_domain\_nyt=1}. A topology of twenty-four square (in |
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gridpoints) tiles, four (2x2) per subdomain, will have |
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\texttt{exch2\_domain\_nxt=12} and \texttt{exch2\_domain\_nyt=2}. |
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Note that these parameters express the tile layout to allow global |
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data files that are tile-layout-neutral and have no bearing on the |
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internal storage of the arrays. The tiles are internally stored in a |
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range from \texttt{1,bi} (in the x axis) and y-axis variable |
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\texttt{bj} is generally ignored within the package. |
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\subsubsection{Arrays Indexed to Tile Number} |
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The following arrays are of size \texttt{NTILES}, are indexed to the |
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tile number, and the indices are omitted in their descriptions. |
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The arrays \varlink{exch2\_tnx}{exch2_tnx} and |
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\varlink{exch2\_tny}{exch2_tny} express the x and y dimensions of each |
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tile. At present for each tile \texttt{exch2\_tnx=sNx} and |
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\texttt{exch2\_tny=sNy}, as assigned in \texttt{SIZE.h}. Future |
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releases of MITgcm are to allow varying tile sizes. |
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The location of the tiles' Cartesian origin within a subdomain are |
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determined by the arrays \varlink{exch2\_tbasex}{exch2_tbasex} and |
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\varlink{exch2\_tbasey}{exch2_tbasey}. These variables are used to |
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relate the location of the edges of the tiles to each other. As an |
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example, in the default six-tile topology (the degenerate case) each |
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index in these arrays are set to 0. The twenty-four, 32x32 cube face |
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case discussed above will have values of 0 or 16, depending on the |
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quadrant the tile falls within the subdomain. The array |
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\varlink{exch2\_myFace}{exch2_myFace} contains the number of the |
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cubeface/subdomain of each tile, numbered 1-6 in the case of the |
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standard cube topology. |
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The arrays \varlink{exch2\_txglobalo}{exch2_txglobalo} and |
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\varlink{exch2\_txglobalo}{exch2_txglobalo} are similar to |
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\varlink{exch2\_tbasex}{exch2_tbasex} and |
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\varlink{exch2\_tbasey}{exch2_tbasey}, but locate the tiles within the |
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global address space, similar to that used by global files. |
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The arrays \varlink{exch2\_isWedge}{exch2_isWedge}, |
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\varlink{exch2\_isEedge}{exch2_isEedge}, |
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\varlink{exch2\_isSedge}{exch2_isSedge}, and |
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\varlink{exch2\_isNedge}{exch2_isNedge} are set to 1 if the indexed |
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tile lies on the edge of a subdomain, 0 if not. The values are used |
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within the topology generator to determine the orientation of |
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neighboring tiles and to indicate whether a tile lies on the corner of |
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a subdomain. The latter case indicates special exchange and numerical |
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handling for the singularities at the eight corners of the cube. |
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\varlink{exch2\_nNeighbours}{exch2_nNeighbours} contains a count of |
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how many neighboring tiles each tile has, and is used for setting |
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bounds for looping over neighboring tiles. |
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\varlink{exch2\_tProc}{exch2_tProc} holds the process rank of each |
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tile, and is used in interprocess communication. |
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\subsubsection{Arrays Indexed to Tile Number and Neighbor} |
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The following arrays are all of size \texttt{MAX\_NEIGHBOURS} $\times$ |
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\texttt{NTILES} and describe the orientations between the the tiles. |
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The array \texttt{exch2\_neighbourId(a,T)} holds the tile number for |
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each of the $n$ neighboring tiles. The neighbor tiles are indexed |
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\texttt{(1,MAX\_NEIGHBOURS} in the order right to left on the north |
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then south edges, and then top to bottom on the east and west edges. |
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Maybe throw in a fig here, eh? |
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The \texttt{exch2\_opposingSend\_record(a,T)} array holds the index c |
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in \texttt{exch2\_neighbourId(b,$T_{n}$)} that holds the tile number T. |
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In other words, |
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\begin{verbatim} |
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exch2_neighbourId( exch2_opposingSend_record(a,T), |
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exch2_neighbourId(a,T) ) = T |
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\end{verbatim} |
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and this provides a back-reference from the neighbor tiles. |
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The arrays \varlink{exch2\_pi}{exch2_pi}, |
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\varlink{exch2\_pj}{exch2_pj}, \varlink{exch2\_oi}{exch2_oi}, |
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\varlink{exch2\_oj}{exch2_oj}, \varlink{exch2\_oi\_f}{exch2_oi_f}, and |
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\varlink{exch2\_oj\_f}{exch2_oj_f} specify the transformations in |
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exchanges between the neighboring tiles. The dimensions of |
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\texttt{exch2\_pi(t,N,T)} and \texttt{exch2\_pj(t,N,T)} are the |
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neighbor ID \textit{N} and the tile number \textit{T} as explained |
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above, plus the transformation vector {\em t }, of length two. The |
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first element of the transformation vector indicates the factor by |
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which variables representing the same vector component of a tile will |
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be multiplied, and the second element indicates the transform to the |
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variable in the other direction. As an example, |
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\texttt{exch2\_pi(1,N,T)} holds the transform of the i-component of a |
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vector variable in tile \texttt{T} to the i-component of tile |
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\texttt{T}'s neighbor \texttt{N}, and \texttt{exch2\_pi(2,N,T)} hold |
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the component of neighbor \texttt{N}'s j-component. |
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Under the current cube topology, one of the two elements of |
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\texttt{exch2\_pi} or \texttt{exch2\_pj} for a given tile \texttt{T} |
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and neighbor \texttt{N} will be 0, reflecting the fact that the vector |
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components are orthogonal. The other element will be 1 or -1, |
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depending on whether the components are indexed in the same or |
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opposite directions. For example, the transform dimension of the |
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arrays for all tile neighbors on the same subdomain will be [1,0], |
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since all tiles on the same subdomain are oriented identically. |
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Vectors that correspond to the orthogonal dimension with the same |
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index direction will have [0,1], whereas those in the opposite index |
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direction will have [0,-1]. |
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{\footnotesize |
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\begin{verbatim} |
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C exch2_pi :: X index row of target to source permutation |
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C :: matrix for each neighbour entry. |
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C exch2_pj :: Y index row of target to source permutation |
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C :: matrix for each neighbour entry. |
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C exch2_oi :: X index element of target to source |
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C :: offset vector for cell-centered quantities |
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C :: of each neighbor entry. |
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C exch2_oj :: Y index element of target to source |
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C :: offset vector for cell-centered quantities |
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C :: of each neighbor entry. |
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C exch2_oi_f :: X index element of target to source |
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C :: offset vector for face quantities |
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C :: of each neighbor entry. |
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C exch2_oj_f :: Y index element of target to source |
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C :: offset vector for face quantities |
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C :: of each neighbor entry. |
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\end{verbatim} |
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} |
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\subsection{Key Routines} |
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\subsection{References} |
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